پديد آورنده :
غني زاده، علي
عنوان :
روش‌هاي تفاضلات متناهي فشرده كارآمد و دقيق براي حل معادلات چند بعدي همرفت-انتشار
مقطع تحصيلي :
كارشناسي ارشد
گرايش تحصيلي :
آناليز عددي
محل تحصيل :
اصفهان : دانشگاه صنعتي اصفهان
صفحه شمار :
نه، 140ص.: مصور ، جدول، نمودار
استاد راهنما :
مهدي تاتاري
توصيفگر ها :
روش فشرده مرتبه‌ي بالا , روش تك بعدي محلي , معادلات همرفت-انتشار , آناليز پايداري فون-نويمان , پايدار نامشروط , فشردگي دقت بالاتر , مقياس دستگاه معادلات جبري , خطاي جداسازي
استاد داور :
رضا مزروعي سبداني، مجيد گازر
تاريخ ورود اطلاعات :
1400/10/29
تاريخ ويرايش اطلاعات :
1400/10/29
چكيده فارسي :
در اين پايان‌نامه دسته‌اي از روش‌هاي فشرده مرتبه‌ي بالا همراه با روش تك بعدي محلي براي حل عددي معادلات همرفت-انتشار چند بعدي مورد توجه قرار گرفته‌اند. اين روش‌ها به دليل فشرده‌گي و دقت زياد پذيرفته شده‌اند، در اين نوع از روش‌ها مشتقات مكاني به صورت ضمني، با الگوي كوچكتر اما با دقت بالاتر تقريب زده مي‌شوند.
راهبرد كاربردي روش تك بعدي محلي در بعد زماني براي كاهش دستگاه معادلات جبري حاصل از روش‌هاي عددي به كار مي‌رود. اين تكنيك با هزينه‌‌ي پائين‌تر باعث مي‌شود مسائل چند بعدي را بتوان برنامه نويسي كرد. براساس تجزيه و تحليل خطاي جداسازي، با يك تغيير جزئي مي‌توان يك روش دقيق‌تر نسبت به روش اصلي بدست آورد.
با رويكرد روش آناليز پايداري فون-نويمان مي توان دريافت كه طرح‌هاي پيشنهادي پايدار نامشروط هستند. برخي نتايج عددي براي اينكه نشان دهند اين طرح‌ها ،كارآمد و دقيق هستند گزارش شده‌اند.
چكيده انگليسي :
In this dissertation, a series of high-order compact finite difference methods combined with local one-dimensional techniques are considered for numerical solution of multidimensional convection-diffusion equations.
In compact difference methodes, spatial derivatives are implicitly approximated with a smaller but higher pattern. These methods are implemented due to their compactness and high accuracy. Derivatives in the epartial differential quation are replaced by numerical differentiation formulas in the equation. In this theseis high-order numerical differentiation are presented. Moreover, in implementing high-order non-compact finite difference methods, problems often occur near the boundary points of the network. By using the present techniques, we use more computational accuracy and efficiency against non-compact methods.
On the other hand, the practical strategy of the local one-dimensional method in the time dimension is used to reduce the system of algebraic equations obtained from numerical methods. This technique allows multidimensional problems to be programmed at a lower cost. Based on the separation error analysis, accuracy of methods is analysed .With the von Neumann stability analysis approach, it can be found that the proposed schemes are unconditionally stable. To solve the problems of multidimensional differential equations, a combination of locally periodic directional methods and locally one-dimensional methods can be used. In these ideas multidimensional equations have been proposed to solve, due to the reduction of the dimensions of the algebraic equations resulting from the discretization of the equations. Computational cost of solving algebraic equations system decreases from order $(N^d)$ to order N, where d and N are spatial dimension and number of network points in each direction is location on the target area respectively. A change in localized method ($ LOD $) improves computational efficiency. The idea of correction of perturbations during separation can be generalized to other separation methods, and additional splitting and separation steps can be avoided so as not to impose additional computational costs on the algebraic equation system.
In Sum, combination of high-order compact finite difference methods with local one-dimensional techniques, makes it possible to achieve the desired accuracy by using fewer points of the network, and less volume of computational operations by reduction of dimensional of the spatial problems.
Some numerical results have been reported to indicate that these schemes are efficient and accurate. The numerical examples provided show the effectiveness of these methods. Implementation of these techniques on other partial differential equations in higher dimensions can be considered as a research problem.
Conclusions:
In this dissertation, high-order intensive methods for solving the convection-diffusion equation in dimensions higher than one have been studied. The ($ LOD $) technique has been used to overcome the high computational cost of solving multidimensional problems. These methods have been shown to be unconditionally stable with low computational cost and high accuracy.
The numerical examples provided show the effectiveness of these methods. Implementation of these techniques on other partial equations in higher dimensions can be considered as a research problem.
استاد راهنما :
مهدي تاتاري
استاد داور :
رضا مزروعي سبداني، مجيد گازر
